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I am working with some field theory and I am at a point where I would like to change my sign convention from $(-,+,+,+)$ to $(+,-,-,-)$. I am worried at some point I will just equations with the wrong sign because I am used to it and just going by memory.

Is there any table with the differences in sign for the "standard" equation (for example Lagrangians, energy-momentum tensor, equation of motion etc?

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    $\begingroup$ I recommend learning how to figure out which convention an equation is using. For example, a Lagrangian has a positive “kinetic” term. $\endgroup$
    – G. Smith
    Commented Oct 27, 2020 at 19:05

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TL;DR: The main rule is that kinetic terms should be positive, cf. above comment by G. Smith.

Example:

  • The Lagrangian density for a scalar field is $${\cal L}~=~\mp \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi -{\cal V}(\phi), $$ with Euler-Lagrange (EL) equation $$ \pm\Box\phi~=~{\cal V}^{\prime}(\phi),$$ if the signature of the Minkowski metric is $(\mp,\pm,\pm,\pm)$, respectively.

    The sign convention for the stress-energy-momentum (SEM) tensor is $$\begin{align} T^{\mu\nu}~=~&\mp \frac{\partial{\cal L}}{\partial (\partial_{\mu}\phi)}\partial^{\nu}\phi \pm g^{\mu\nu}{\cal L} ~=~\partial^{\mu}\phi\partial^{\nu}\phi \pm g^{\mu\nu}{\cal L}, \cr T_{\mu\nu}~=~&\mp \frac{\partial{\cal L}}{\partial (\partial^{\mu}\phi)}\partial_{\nu}\phi \pm g_{\mu\nu}{\cal L} ~=~\partial_{\mu}\phi\partial_{\nu}\phi \pm g_{\mu\nu}{\cal L}, \end{align}$$ cf. this related Phys.SE post.

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Check Appendix E of Burgess & Moore's The Standard Model: A Primer. There is a pretty good collection of results like that there.

PS. Sorry to lose another soldier to the dark side that is the $(+---)$ metric (just kidding)

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  • $\begingroup$ Jokes aside, (+−−−) is quite ugly when one wants to check some non relativistic limit, no doubt that the "mostly plus" is a more "natural" choice :D Here a short list of books listed by convention en.wikipedia.org/wiki/Sign_convention $\endgroup$
    – Quillo
    Commented Nov 7, 2023 at 11:30

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